A Maximal Element Theorem in FWC-Spaces and Its Applications

Abstract

A maximal element theorem is proved in finite weakly convex spaces (FWC-spaces, in short) which have no linear, convex, and topological structure. Using the maximal element theorem, we develop new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in FWC-spaces. The results represented in this paper unify and extend some known results in the literature.

1. Introduction and Preliminaries

In 1983, by using fixed point theorems for set-valued mappings, Yannelis and Prabhakar [1] proved three existence theorems of maximal elements under the setting of locally convex topological vector spaces. In 1985, Yannelis [2] improved the Fan-Browder-type fixed point theorem and obtained an existence result of maximal elements by using this fixed point theorem. Since then, many maximal element theorems and their applications have been established in the setting of topological vector spaces; see, for example, [3–8] and the references therein.

It is well known that the linearity and convexity assumptions play crucial roles in most of the known existence results of maximal elements, which strictly restrict the applicable range of these maximal element theorems. Considering this fact, Zhang and Wu [9] proved an existence theorem of maximal elements in noncompact -spaces and obtained some minimax inequalities, variational inequalities, and quasivariational inequalities by using this maximal element theorem. Subsequently, Wu [10] used existence theorems of maximal elements to prove equilibrium existence theorems for qualitative games and abstract economies in noncompact -spaces. Recently, by using a generalization of the Fan-Browder fixed point theorem, Balaj and Lin [11] proved a new fixed point theorem for set-valued mappings in -convex spaces from which they derived several coincidence theorems and existence theorems for maximal elements. As applications, they obtained some existence theorems of solutions to the generalized equilibrium problem and minimax problem.

Motivated and inspired by the work mentioned above, in this paper, we prove a new maximal element theorem in -spaces (see Definition 1) without any linear, convex, and topological structure. As applications of this theorem, we obtain some new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in -spaces.

Now, we introduce some notation and definitions. For a nonempty set , and denote the family of all subsets of and the family of nonempty finite subsets of , respectively. For every , denotes the cardinality of . If is a topological space, then denotes the closure of . If is a vector space, then we denote by the convex hull of . Let be a set-valued mapping with being a nonempty set. We define the mapping by for each . If is a topological space, we say that is compact if is compact. If and are both topological spaces, we say that is upper semicontinuous (resp., lower semicontinuous) if for every closed subset of , the set (resp., ) is closed. Let denote the standard -dimensional simplex with vertices . For a nonempty subset , let denote the convex hull of the vertices .

A nonempty topological space is contractible if the identity mapping on is homotopic to a constant mapping. Every nonempty convex subset of a topological vector space is contractible, but the converse is not true in general. A subset of a topological space is called to be compactly closed (resp., compactly open) in if for each nonempty compact subset of , is closed (resp., open) in . The notions of compactly closed (resp., compactly open) sets are true generalizations of closed (resp., open) sets. Note that there exists a nonempty subset of the topological vector space such that for each nonempty compact subset of , is closed in , but is not closed. For details, see Kelley [12, page 240] or Wilansky [13, page 143].

Let be a topological vector space. For every , let us define a continuous mapping by for each . This mapping motivates us to introduce an abstract convex space which does not possess any linear, convex, and topological structure and is described in the following definition.

Definition 1 (see [14]). A triple is said to be a finite weakly convex space (-space, in short) if , are two nonempty sets and for each where some elements in may be same, there exists a set-valued mapping with nonempty values. When , the space is denoted by . In case , let . Let and . is said to be an -subspace of relative to if for each and for each , we have , where . We note that if is nonempty and is an -subspace of relative to , then is automatically nonempty. When , is said to be an -subspace of .
It is worthwhile noticing that and in Definition 1 do not possess any linear, convex, and topological structure. Major examples of -spaces are convex subsets of topological vector spaces, hyperconvex metric spaces introduced by Aronszajn and Panitchpakdi [15], Lassonde’s convex spaces in [16], -spaces introduced by Horvath [17], -convex spaces introduced by Park and Kim [18], -convex spaces introduced by Ben-El-Mechaiekh et al. [19], --spaces introduced by Verma [20–22], pseudo--spaces introduced by Lai et al. [23], -spaces due to Khanh et al. [24], -spaces due to Ding [25], and many other topological spaces with abstract convex structure (see, e.g., [26] and the references therein).
Taking Lassonde’s convex space, -space, and hyperconvex metric space as examples, we show that these three spaces are particular forms of -spaces. Let be a convex space in [16]; that is, a nonempty convex set in a vector space with any topology that induces the Euclidean topology on the convex hulls of its finite subsets. Then for every , define a continuous mapping as follows: Therefore, forms an -space. Let be an -space in [17], where is a family of nonempty contractible subsets of indexed by such that whenever . Then by Theorem 1 of Horvath [17], for every , there exists a continuous mapping and thus, is an -space. Let be a hyperconvex metric space in [15]. Then by Lemma 2.3 of Yuan [27], we know that is an -space and thus, is an -space.

Definition 2. Let be an -space and a topological space. The class of better admissible mappings is defined as follows: a set-valued mapping belongs to if and only if for every and for every continuous mapping , the composition has a fixed point. When , we will write instead of .

Remark 3. Since and in Definition 2 are nonempty sets which do not possess any linear, convex, and topological structure, the class includes many important classes of mappings as special cases, for example, the class of Kakutani's mappings (i.e., the upper semicontinuous set-valued mappings with nonempty compact convex values and codomain being convex set in a topological vector space), the class in Park and Kim [18], the class in Ben-El-Mechaiekh et al. [19], and the class in Ding [25].

Example 4. Let and , where is a family of subsets of . We can verify that is not a topological space. For simplicity, we write instead of . Let with the Euclidean metric topology. For each , define a set-valued mapping by for each . It is easy to see that forms an -space. Now we define a set-valued mapping by Then for each , we have . Therefore, the composition is an upper semicontinuous set-valued mapping with nonempty compact contractible values. By Lemma 1 of [28], for every continuous function , the composition has a fixed point. Therefore, .

Lemma 5. Let be an index set. For each , let be an -space. Let , , and . Then is also an -space.

Proof. For each , let be the projection of onto . For every , let . Since each is an -space, it follows that there exists a set-valued mapping with nonempty values. Define a set-valued mapping by It is clear that has nonempty values. Therefore, is also an -space.

2. A Maximal Element Theorem

Our first result is the following maximal element theorem.

Theorem 6. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let , , , and be set-valued mappings such that(i)for each , ;(ii)for each , is compactly open;(iii)for each and each , (iv)one of the following conditions holds:(iv₁)there exists such that and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that .

Proof. We prove Theorem 6 distinguishing the following two cases.
Case (iv₁). Assume (iv₁) holds. Suppose that the conclusion of Theorem 6 does not hold. Then for each , and hence, there exists ; that is, . Therefore, we have which implies that . Since is compact and each is compactly open by (ii), it follows that there exists such that By the first part of (iv₁), we have Then it follows from (7) and (8) that where . By the definition of -spaces, there exists a set-valued mapping with nonempty values. By the second part of (iv₁), is compact subset of . By (9), we have and thus, ; that is, is an open cover of the compact set . Let be the partition of unity subordinated to this cover and then define a mapping by for each . Clearly, is continuous and for each , we have where is defined by . Then we have Define a set-valued mapping as follows: Now, we show that for each , is an -subspace of relative to . In fact, let and . Then . By (iii), ; that is, . Therefore, we have . By (i), we know that , which implies that for each , is an -subspace of relative to . Hence, by (12) and (13), we have This shows that
On the other hand, since , it follows that the composition mapping has a fixed point ; that is, . Let such that . Choose such that Then by (13) and (16), we have which contradicts (15). Thus, there must exist a point such that .
Case (iv₂). Assume (iv₂) holds. Suppose that the conclusion of Theorem 6 is not true. Then by using the same method as in Case (iv₁), we have which implies that . By (ii), we know that is a family of closed sets in . Thus, there exists such that that is, . By (iv₂), there exists a subset of containing such that is an -subspace of relative to and Therefore, we have
Since , it follows that . Therefore, we have Since is a compact subset of , it follows from (ii) and (22) that there exists such that By the fact that is an -subspace of relative to , we can see that the triple is also an -space. Hence, there exists a set-valued mapping with nonempty values. Assume that is the partition of unity subordinated to the open cover . Then for every , we have Furthermore, we define a continuous mapping by for each . Let . Since , it follows that . Then the composition has a fixed point ; that is, . Let such that . Then we have where . Let . By (i) and (iii), for each and each , we have thus, we have the following: Hence, there exists such that . On the other hand, by the definitions of and of the partition , we have which is a contradiction. Therefore, the conclusion of Theorem 6 holds.

Remark 7. (iii) of Theorem 6 can be replaced by the following equivalent condition:
for each , is an -subspace of relative to .

Proof. (iii) : let and . Then . Thus, . Since by (iii) of Theorem 6, it follows that ; that is, . Therefore, we have , which implies that holds.
(iii): let , , and . Then there exists such that . Therefore, we get This means that . By , we have and hence, . Let . Then , which implies that

Example 8. Let be endowed with the Euclidean topology. Let and , where is a family of subsets of . It is easy to check that is not a topological space. For simplicity, we will write instead of . Define a set-valued mappings such that which is open in . Therefore, is compactly open-valued, and hence, (ii) of Theorem 6 is satisfied. Furthermore, define a set-valued mapping such that Now, let be defined by Then we have , which is compact subset of . Let . Then (iv₁) of Theorem 6 is satisfied automatically. In order to check (iii) of Theorem 6, for every , we define a set-valued mapping by for all . Then forms an -space. For each , we have We can see that for each , is an -subspace of relative to . Therefore, by Remark 7, we know that (iii) of Theorem 6 holds. Finally, we show that, for each , and . By the definition of , we can obtain as follows: Thus, we can easily see that for each . Hence, for each , . By using the same method as in Example 4, we can prove that . Therefore, all the hypotheses of Theorem 6 are satisfied. We can see that there exists a point such that .

If in Theorem 6, then Theorem 6 deduces the following result.

Corollary 9. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let , , and be set-valued mappings such that(i)for each , is compactly open;(ii)for each and each , (iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that .

Taking and in Corollary 9, we can obtain the following result.

Corollary 10. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let and be set-valued mappings such that(i)for each , is compactly open;(ii)for each and each , (iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and
Then there exists such that .

3. Existence of Solutions to Variational Relation Problem

In 2008, Luc [29] introduced a variational relation problem which unifies many equilibrium problems, optimization problems, and variational or differential inclusion problems. Since then, further studies on variational relation problems were investigated by many authors; see, for example, [30–32] and the references therein.

Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a set-valued mapping. In this section, we will study the following variational relation problems in -spaces.(1)Let be a relation linking and . Find such that holds for each .(2)Let be a nonempty set, a set-valued mapping, and a relation linking and . Find such that holds for each and each .

By applying Corollary 10, we have the following existence theorem of solutions to the variational relation problem in -spaces.

Theorem 11. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let be a set-valued mapping and let be a relation linking elements , such that(i)for each , the set is compactly open;(ii)for each , each and each , there exists such that holds;(iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and
Then there exists such that holds for each .

Proof. Define a set-valued mapping by By (i), for each , is compactly open. By (ii), for each and each , we have which implies that By (iii), we know that one of the following conditions holds:(a)there exists such that and for each , is a compact subset of ;(b)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and Therefore, by Corollary 10, there exists such that ; that is, holds for each .

By taking and for every in Theorem 11, we can obtain the following result.

Corollary 12. Let be an FWC-space and a nonempty compact subset of , where is a Hausdorff topological space. Let , where is the identity mapping on . Let be a relation linking elements , such that(i)for each , the set is compactly open;(ii)for each , each and each , there exists such that holds;(iii)one of the following conditions holds:(iii₁)there exists such that and is a compact set-valued mapping for each ;(iii₂)for each , there exists a compact subset of containing such that is an -subspace of and
Then there exists such that holds for each .

Corollary 13. Let be an FWC-space, where is a Hausdorff compact topological space. Let , where is the identity mapping on . Let be a relation linking elements , such that the following conditions hold:(i)for each , the set is compactly open;(ii)for each , each and each , there exists such that holds.
Then there exists such that holds for each .

Proof. Let . Then (iii₁) of Corollary 12 is satisfied automatically. Hence, the conclusion of Corollary 13 follows from Corollary 12.

Remark 14. It is interesting to compare Corollary 13 with Theorem 2.1 of Pu and Yang [32] in the following aspects: (i) of Corollary 13 is weaker than (i) of Theorem 2.1 of Pu and Yang [32], which can be stated as follows: for each , the set is closed; (ii) of Theorem 2.1 of Pu and Yang [32] can be stated as follows: for each , there exists a continuous mapping such that, for each , there exists such that holds, where . By (ii) of Theorem 2.1 of Pu and Yang [32], we know that in Theorem 2.1 of Pu and Yang [32] forms an -space; in Corollary 13, the topological space needs not to have the fixed point property, but in Theorem 2.1 of Pu and Yang [32] needs to possess the fixed point property; for the identity mapping on in Theorem 2.1 of Pu and Yang [32], we must have . In fact, for every and for every continuous mapping , the composition is continuous, where coincides with the one in (ii) of Theorem 2.1 of Pu and Yang [32]. Then by Brouwer fixed point theorem, there exists such that , which implies that .

Theorem 15. Let be an FWC-space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , be set-valued mappings and a relation linking elements , . Assume that(i)for each , the set is compactly open, where is defined by for each ;(ii)for each , each and each , there exists such that holds for each ;(iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and
Then there exists such that holds for each and each .

Proof. Let the relation on and be defined by holds if and only if . Then by (i), for each , the set is compactly open. By (ii), for each , each and each , there exists such that that is, . Thus, holds. By (iii), we know that one of the following conditions holds:(a)there exists such that and for each , is compact subset of ;(b)for each , there exists a subset of containing such that is an -subspace of and where is a compact subset of . Therefore, by Theorem 11, there exists such that holds for each ; that is, holds for each and each .

Remark 16. Theorem 15 generalizes Theorem 3.1 of Balaj and Lin [30] in the following aspects: (a) The underlying spaces of Theorem 15 and Theorem 3.1 of Balaj and Lin [30] are -spaces and convex spaces, respectively. It follows from the previous analysis that -spaces include convex spaces as special cases; (b) The class of better admissible mappings in Theorem 15 and Theorem 3.1 of Balaj and Lin [30] are and , respectively. By Remark 3, we know that is contained in ; (c) in Theorem 15 does not possess any topological structure; (d) in Theorem 15 needs not to be compact. In fact, if in Theorem 15 is compact, then we know that (iii₁) of Theorem 15 is satisfied by taking ; (e) (ii) of Theorem 3.1 of Balaj and Lin [30] can be stated as follows: for each , each and each , there exists such that for all . (ii) of Theorem 15 is weaker than (ii) of Theorem 3.1 of Balaj and Lin [30]. In fact, for every , we can define a continuous mapping by Therefore, forms an -space. On the basis of this fact, we can see that (ii) of Theorem 3.1 of Balaj and Lin [30] implies (ii) of Theorem 15.
(2) Theorem 15 is equivalent to Theorem 11. In fact, we have shown that Theorem 11 implies Theorem 15. Conversely, if and for each in Theorem 15, then Theorem 15 becomes Theorem 11.

Corollary 17. Let be defined by for each . Theorem 15 is true if (i) of Theorem 15 is replaced by one of the following conditions: for each , the set is compactly closed; is a topological space, the set-valued mapping is lower semicontinuous, and has open values.

Proof. Suppose that is satisfied. Then is compactly closed for each . Thus, is compactly open. If holds, then by the definition of a lower semicontinuous set-valued mapping, for each , the set is open and thus, compactly open.

4. Generalized Equilibrium Theorems

In recent years, many authors (see, e.g., [33–35] and the references therein) studied one or more of the following generalized equilibrium problems.

Let and be nonempty sets and a topological space. Let and be set-valued mappings. Find such that one of the following situations occurs:

Let be another nonempty set, a set-valued mapping, and a single-valued mapping. The generalized implicit vector equilibrium problem is to find such that, for each , there exists satisfying . For more details, the reader may consult [35] and the references therein.

In this section, as applications of Theorem 6, we will prove new existence theorems of solutions to generalized equilibrium problems in -spaces without any linear, convex, and topological structure.

Theorem 18. Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , , , , and be set-valued mappings such that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Proof. Define and by